\(\int \frac {(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}}{(d+e x)^{7/2}} \, dx\) [2256]

Optimal result

Integrand size = 46, antiderivative size = 316 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\frac {2 (2 c d-b e)^2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}+\frac {2 (2 c d-b e) (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (d+e x)^{5/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e^2 (d+e x)^{7/2}}-\frac {2 (2 c d-b e)^{5/2} (e f-d g) \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{e^2} \]

[Out]

2/3*(-b*e+2*c*d)*(-d*g+e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/e^2/(e*x+d)^(3/2)+2/5*(-d*g+e*f)*(d*(-b*e+c
*d)-b*e^2*x-c*e^2*x^2)^(5/2)/e^2/(e*x+d)^(5/2)-2/7*g*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(7/2)/c/e^2/(e*x+d)^(7/2
)-2*(-b*e+2*c*d)^(5/2)*(-d*g+e*f)*arctanh((d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/(-b*e+2*c*d)^(1/2)/(e*x+d)^(1
/2))/e^2+2*(-b*e+2*c*d)^2*(-d*g+e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/e^2/(e*x+d)^(1/2)

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {808, 678, 674, 214} \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=-\frac {2 (2 c d-b e)^{5/2} (e f-d g) \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{e^2}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (d+e x)^{5/2}}+\frac {2 (2 c d-b e) (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}+\frac {2 (2 c d-b e)^2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e^2 (d+e x)^{7/2}} \]

[In]

Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^(7/2),x]

[Out]

(2*(2*c*d - b*e)^2*(e*f - d*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(e^2*Sqrt[d + e*x]) + (2*(2*c*d - b*
e)*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(3*e^2*(d + e*x)^(3/2)) + (2*(e*f - d*g)*(d*(c*d -
b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(5*e^2*(d + e*x)^(5/2)) - (2*g*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2)
)/(7*c*e^2*(d + e*x)^(7/2)) - (2*(2*c*d - b*e)^(5/2)*(e*f - d*g)*ArcTanh[Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*
x^2]/(Sqrt[2*c*d - b*e]*Sqrt[d + e*x])])/e^2

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 674

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[2*e, Subst[Int[1/(
2*c*d - b*e + e^2*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^
2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 678

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((
a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x] - Dist[p*((2*c*d - b*e)/(e^2*(m + 2*p + 1))), Int[(d + e*x)^(m + 1)*
(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a
*e^2, 0] && GtQ[p, 0] && (LeQ[-2, m, 0] || EqQ[m + p + 1, 0]) && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p]

Rule 808

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Dist[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)
*(2*c*f - b*g))/(c*e*(m + 2*p + 2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g
, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[m + 2*p + 2, 0] && (NeQ[m, 2] || Eq
Q[d, 0])

Rubi steps \begin{align*} \text {integral}& = -\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e^2 (d+e x)^{7/2}}-\frac {\left (2 \left (\frac {7}{2} e \left (-2 c e^2 f+b e^2 g\right )-\frac {7}{2} \left (-c e^3 f+\left (-c d e^2+b e^3\right ) g\right )\right )\right ) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx}{7 c e^3} \\ & = \frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (d+e x)^{5/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e^2 (d+e x)^{7/2}}+\frac {((2 c d-b e) (e f-d g)) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx}{e} \\ & = \frac {2 (2 c d-b e) (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (d+e x)^{5/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e^2 (d+e x)^{7/2}}+\frac {\left ((2 c d-b e)^2 (e f-d g)\right ) \int \frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{3/2}} \, dx}{e} \\ & = \frac {2 (2 c d-b e)^2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}+\frac {2 (2 c d-b e) (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (d+e x)^{5/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e^2 (d+e x)^{7/2}}+\frac {\left ((2 c d-b e)^3 (e f-d g)\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{e} \\ & = \frac {2 (2 c d-b e)^2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}+\frac {2 (2 c d-b e) (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (d+e x)^{5/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e^2 (d+e x)^{7/2}}+\left (2 (2 c d-b e)^3 (e f-d g)\right ) \text {Subst}\left (\int \frac {1}{-2 c d e^2+b e^3+e^2 x^2} \, dx,x,\frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{\sqrt {d+e x}}\right ) \\ & = \frac {2 (2 c d-b e)^2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}+\frac {2 (2 c d-b e) (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (d+e x)^{5/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e^2 (d+e x)^{7/2}}-\frac {2 (2 c d-b e)^{5/2} (e f-d g) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{e^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.80 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\frac {2 ((d+e x) (-b e+c (d-e x)))^{5/2} \left (\frac {15 b^3 e^3 g+b^2 c e^2 (161 e f-206 d g+45 e g x)+c^3 \left (-526 d^3 g+3 e^3 x^2 (7 f+5 g x)-2 d e^2 x (56 f+33 g x)+d^2 e (511 f+157 g x)\right )+b c^2 e \left (612 d^2 g+e^2 x (77 f+45 g x)-d e (567 f+167 g x)\right )}{c (-c d+b e+c e x)^2}+\frac {105 (-2 c d+b e)^{5/2} (-e f+d g) \arctan \left (\frac {\sqrt {-b e+c (d-e x)}}{\sqrt {-2 c d+b e}}\right )}{(-b e+c (d-e x))^{5/2}}\right )}{105 e^2 (d+e x)^{5/2}} \]

[In]

Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^(7/2),x]

[Out]

(2*((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2)*((15*b^3*e^3*g + b^2*c*e^2*(161*e*f - 206*d*g + 45*e*g*x) + c^3*(-
526*d^3*g + 3*e^3*x^2*(7*f + 5*g*x) - 2*d*e^2*x*(56*f + 33*g*x) + d^2*e*(511*f + 157*g*x)) + b*c^2*e*(612*d^2*
g + e^2*x*(77*f + 45*g*x) - d*e*(567*f + 167*g*x)))/(c*(-(c*d) + b*e + c*e*x)^2) + (105*(-2*c*d + b*e)^(5/2)*(
-(e*f) + d*g)*ArcTan[Sqrt[-(b*e) + c*(d - e*x)]/Sqrt[-2*c*d + b*e]])/(-(b*e) + c*(d - e*x))^(5/2)))/(105*e^2*(
d + e*x)^(5/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(947\) vs. \(2(286)=572\).

Time = 0.36 (sec) , antiderivative size = 948, normalized size of antiderivative = 3.00

method	result	size
default	\(\frac {2 \sqrt {-\left (e x +d \right ) \left (x c e +b e -c d \right )}\, \left (15 c^{3} e^{3} g \,x^{3} \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}+45 b \,c^{2} e^{3} g \,x^{2} \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}-66 c^{3} d \,e^{2} g \,x^{2} \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}+21 c^{3} e^{3} f \,x^{2} \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}+105 \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) b^{3} c d \,e^{3} g -105 \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) b^{3} c \,e^{4} f -630 \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) b^{2} c^{2} d^{2} e^{2} g +630 \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) b^{2} c^{2} d \,e^{3} f +1260 \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) b \,c^{3} d^{3} e g -1260 \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) b \,c^{3} d^{2} e^{2} f -840 \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{4} d^{4} g +840 \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{4} d^{3} e f +45 b^{2} c \,e^{3} g x \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}-167 b \,c^{2} d \,e^{2} g x \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}+77 b \,c^{2} e^{3} f x \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}+157 c^{3} d^{2} e g x \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}-112 c^{3} d \,e^{2} f x \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}+15 \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}\, b^{3} e^{3} g -206 \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}\, b^{2} c d \,e^{2} g +161 \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}\, b^{2} c \,e^{3} f +612 \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}\, b \,c^{2} d^{2} e g -567 \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}\, b \,c^{2} d \,e^{2} f -526 \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}\, c^{3} d^{3} g +511 \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}\, c^{3} d^{2} e f \right )}{105 \sqrt {e x +d}\, \sqrt {-x c e -b e +c d}\, e^{2} c \sqrt {b e -2 c d}}\)	\(948\)

[In]

int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(7/2),x,method=_RETURNVERBOSE)

[Out]

2/105*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(15*c^3*e^3*g*x^3*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+45*b*c^2*e^3
*g*x^2*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-66*c^3*d*e^2*g*x^2*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+21
*c^3*e^3*f*x^2*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+105*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b
^3*c*d*e^3*g-105*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b^3*c*e^4*f-630*arctan((-c*e*x-b*e+c*d)^(1/2
)/(b*e-2*c*d)^(1/2))*b^2*c^2*d^2*e^2*g+630*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b^2*c^2*d*e^3*f+12
60*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^3*d^3*e*g-1260*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*
d)^(1/2))*b*c^3*d^2*e^2*f-840*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^4*d^4*g+840*arctan((-c*e*x-b*
e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^4*d^3*e*f+45*b^2*c*e^3*g*x*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-167*b*c^
2*d*e^2*g*x*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+77*b*c^2*e^3*f*x*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)
+157*c^3*d^2*e*g*x*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-112*c^3*d*e^2*f*x*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*
d)^(1/2)+15*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*b^3*e^3*g-206*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*b^
2*c*d*e^2*g+161*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*b^2*c*e^3*f+612*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1
/2)*b*c^2*d^2*e*g-567*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*b*c^2*d*e^2*f-526*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2
*c*d)^(1/2)*c^3*d^3*g+511*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*c^3*d^2*e*f)/(e*x+d)^(1/2)/(-c*e*x-b*e+c*d)
^(1/2)/e^2/c/(b*e-2*c*d)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 949, normalized size of antiderivative = 3.00 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\left [-\frac {105 \, \sqrt {2 \, c d - b e} {\left ({\left (4 \, c^{3} d^{3} e - 4 \, b c^{2} d^{2} e^{2} + b^{2} c d e^{3}\right )} f - {\left (4 \, c^{3} d^{4} - 4 \, b c^{2} d^{3} e + b^{2} c d^{2} e^{2}\right )} g + {\left ({\left (4 \, c^{3} d^{2} e^{2} - 4 \, b c^{2} d e^{3} + b^{2} c e^{4}\right )} f - {\left (4 \, c^{3} d^{3} e - 4 \, b c^{2} d^{2} e^{2} + b^{2} c d e^{3}\right )} g\right )} x\right )} \log \left (-\frac {c e^{2} x^{2} - 3 \, c d^{2} + 2 \, b d e - 2 \, {\left (c d e - b e^{2}\right )} x - 2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {2 \, c d - b e} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \, {\left (15 \, c^{3} e^{3} g x^{3} + 3 \, {\left (7 \, c^{3} e^{3} f - {\left (22 \, c^{3} d e^{2} - 15 \, b c^{2} e^{3}\right )} g\right )} x^{2} + 7 \, {\left (73 \, c^{3} d^{2} e - 81 \, b c^{2} d e^{2} + 23 \, b^{2} c e^{3}\right )} f - {\left (526 \, c^{3} d^{3} - 612 \, b c^{2} d^{2} e + 206 \, b^{2} c d e^{2} - 15 \, b^{3} e^{3}\right )} g - {\left (7 \, {\left (16 \, c^{3} d e^{2} - 11 \, b c^{2} e^{3}\right )} f - {\left (157 \, c^{3} d^{2} e - 167 \, b c^{2} d e^{2} + 45 \, b^{2} c e^{3}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{105 \, {\left (c e^{3} x + c d e^{2}\right )}}, -\frac {2 \, {\left (105 \, \sqrt {-2 \, c d + b e} {\left ({\left (4 \, c^{3} d^{3} e - 4 \, b c^{2} d^{2} e^{2} + b^{2} c d e^{3}\right )} f - {\left (4 \, c^{3} d^{4} - 4 \, b c^{2} d^{3} e + b^{2} c d^{2} e^{2}\right )} g + {\left ({\left (4 \, c^{3} d^{2} e^{2} - 4 \, b c^{2} d e^{3} + b^{2} c e^{4}\right )} f - {\left (4 \, c^{3} d^{3} e - 4 \, b c^{2} d^{2} e^{2} + b^{2} c d e^{3}\right )} g\right )} x\right )} \arctan \left (\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {-2 \, c d + b e} \sqrt {e x + d}}{c e^{2} x^{2} + b e^{2} x - c d^{2} + b d e}\right ) - {\left (15 \, c^{3} e^{3} g x^{3} + 3 \, {\left (7 \, c^{3} e^{3} f - {\left (22 \, c^{3} d e^{2} - 15 \, b c^{2} e^{3}\right )} g\right )} x^{2} + 7 \, {\left (73 \, c^{3} d^{2} e - 81 \, b c^{2} d e^{2} + 23 \, b^{2} c e^{3}\right )} f - {\left (526 \, c^{3} d^{3} - 612 \, b c^{2} d^{2} e + 206 \, b^{2} c d e^{2} - 15 \, b^{3} e^{3}\right )} g - {\left (7 \, {\left (16 \, c^{3} d e^{2} - 11 \, b c^{2} e^{3}\right )} f - {\left (157 \, c^{3} d^{2} e - 167 \, b c^{2} d e^{2} + 45 \, b^{2} c e^{3}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}\right )}}{105 \, {\left (c e^{3} x + c d e^{2}\right )}}\right ] \]

[In]

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(7/2),x, algorithm="fricas")

[Out]

[-1/105*(105*sqrt(2*c*d - b*e)*((4*c^3*d^3*e - 4*b*c^2*d^2*e^2 + b^2*c*d*e^3)*f - (4*c^3*d^4 - 4*b*c^2*d^3*e +
b^2*c*d^2*e^2)*g + ((4*c^3*d^2*e^2 - 4*b*c^2*d*e^3 + b^2*c*e^4)*f - (4*c^3*d^3*e - 4*b*c^2*d^2*e^2 + b^2*c*d*
e^3)*g)*x)*log(-(c*e^2*x^2 - 3*c*d^2 + 2*b*d*e - 2*(c*d*e - b*e^2)*x - 2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b
*d*e)*sqrt(2*c*d - b*e)*sqrt(e*x + d))/(e^2*x^2 + 2*d*e*x + d^2)) - 2*(15*c^3*e^3*g*x^3 + 3*(7*c^3*e^3*f - (22
*c^3*d*e^2 - 15*b*c^2*e^3)*g)*x^2 + 7*(73*c^3*d^2*e - 81*b*c^2*d*e^2 + 23*b^2*c*e^3)*f - (526*c^3*d^3 - 612*b*
c^2*d^2*e + 206*b^2*c*d*e^2 - 15*b^3*e^3)*g - (7*(16*c^3*d*e^2 - 11*b*c^2*e^3)*f - (157*c^3*d^2*e - 167*b*c^2*
d*e^2 + 45*b^2*c*e^3)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d))/(c*e^3*x + c*d*e^2), -2/
105*(105*sqrt(-2*c*d + b*e)*((4*c^3*d^3*e - 4*b*c^2*d^2*e^2 + b^2*c*d*e^3)*f - (4*c^3*d^4 - 4*b*c^2*d^3*e + b^
2*c*d^2*e^2)*g + ((4*c^3*d^2*e^2 - 4*b*c^2*d*e^3 + b^2*c*e^4)*f - (4*c^3*d^3*e - 4*b*c^2*d^2*e^2 + b^2*c*d*e^3
)*g)*x)*arctan(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(-2*c*d + b*e)*sqrt(e*x + d)/(c*e^2*x^2 + b*e^2*
x - c*d^2 + b*d*e)) - (15*c^3*e^3*g*x^3 + 3*(7*c^3*e^3*f - (22*c^3*d*e^2 - 15*b*c^2*e^3)*g)*x^2 + 7*(73*c^3*d^
2*e - 81*b*c^2*d*e^2 + 23*b^2*c*e^3)*f - (526*c^3*d^3 - 612*b*c^2*d^2*e + 206*b^2*c*d*e^2 - 15*b^3*e^3)*g - (7
*(16*c^3*d*e^2 - 11*b*c^2*e^3)*f - (157*c^3*d^2*e - 167*b*c^2*d*e^2 + 45*b^2*c*e^3)*g)*x)*sqrt(-c*e^2*x^2 - b*
e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d))/(c*e^3*x + c*d*e^2)]

Sympy [F]

\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\int \frac {\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {5}{2}} \left (f + g x\right )}{\left (d + e x\right )^{\frac {7}{2}}}\, dx \]

[In]

integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2)/(e*x+d)**(7/2),x)

[Out]

Integral((-(d + e*x)*(b*e - c*d + c*e*x))**(5/2)*(f + g*x)/(d + e*x)**(7/2), x)

Maxima [F]

\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\int { \frac {{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {5}{2}} {\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(7/2),x, algorithm="maxima")

[Out]

integrate((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*(g*x + f)/(e*x + d)^(7/2), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1074 vs. \(2 (286) = 572\).

Time = 0.36 (sec) , antiderivative size = 1074, normalized size of antiderivative = 3.40 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\frac {2 \, {\left (\frac {105 \, {\left (8 \, c^{3} d^{3} e f - 12 \, b c^{2} d^{2} e^{2} f + 6 \, b^{2} c d e^{3} f - b^{3} e^{4} f - 8 \, c^{3} d^{4} g + 12 \, b c^{2} d^{3} e g - 6 \, b^{2} c d^{2} e^{2} g + b^{3} d e^{3} g\right )} \arctan \left (\frac {\sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right )}{\sqrt {-2 \, c d + b e}} + \frac {420 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{9} d^{2} e f - 420 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{8} d e^{2} f + 105 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{2} c^{7} e^{3} f - 420 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{9} d^{3} g + 420 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{8} d^{2} e g - 105 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{2} c^{7} d e^{2} g + 70 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{8} d e f - 35 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c^{7} e^{2} f - 70 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{8} d^{2} g + 35 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c^{7} d e g + 21 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )}^{2} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{7} e f - 21 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )}^{2} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{7} d g + 15 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )}^{3} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{6} g}{c^{7}}\right )}}{105 \, e^{2}} - \frac {2 \, {\left (840 \, c^{4} d^{3} e f \arctan \left (\frac {\sqrt {2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right ) - 1260 \, b c^{3} d^{2} e^{2} f \arctan \left (\frac {\sqrt {2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right ) + 630 \, b^{2} c^{2} d e^{3} f \arctan \left (\frac {\sqrt {2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right ) - 105 \, b^{3} c e^{4} f \arctan \left (\frac {\sqrt {2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right ) - 840 \, c^{4} d^{4} g \arctan \left (\frac {\sqrt {2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right ) + 1260 \, b c^{3} d^{3} e g \arctan \left (\frac {\sqrt {2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right ) - 630 \, b^{2} c^{2} d^{2} e^{2} g \arctan \left (\frac {\sqrt {2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right ) + 105 \, b^{3} c d e^{3} g \arctan \left (\frac {\sqrt {2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right ) + 644 \, \sqrt {2 \, c d - b e} \sqrt {-2 \, c d + b e} c^{3} d^{2} e f - 644 \, \sqrt {2 \, c d - b e} \sqrt {-2 \, c d + b e} b c^{2} d e^{2} f + 161 \, \sqrt {2 \, c d - b e} \sqrt {-2 \, c d + b e} b^{2} c e^{3} f - 764 \, \sqrt {2 \, c d - b e} \sqrt {-2 \, c d + b e} c^{3} d^{3} g + 824 \, \sqrt {2 \, c d - b e} \sqrt {-2 \, c d + b e} b c^{2} d^{2} e g - 251 \, \sqrt {2 \, c d - b e} \sqrt {-2 \, c d + b e} b^{2} c d e^{2} g + 15 \, \sqrt {2 \, c d - b e} \sqrt {-2 \, c d + b e} b^{3} e^{3} g\right )}}{105 \, \sqrt {-2 \, c d + b e} c e^{2}} \]

[In]

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(7/2),x, algorithm="giac")

[Out]

2/105*(105*(8*c^3*d^3*e*f - 12*b*c^2*d^2*e^2*f + 6*b^2*c*d*e^3*f - b^3*e^4*f - 8*c^3*d^4*g + 12*b*c^2*d^3*e*g
- 6*b^2*c*d^2*e^2*g + b^3*d*e^3*g)*arctan(sqrt(-(e*x + d)*c + 2*c*d - b*e)/sqrt(-2*c*d + b*e))/sqrt(-2*c*d + b
*e) + (420*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^9*d^2*e*f - 420*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c^8*d*e^2*f +
105*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^2*c^7*e^3*f - 420*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^9*d^3*g + 420*sqr
t(-(e*x + d)*c + 2*c*d - b*e)*b*c^8*d^2*e*g - 105*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^2*c^7*d*e^2*g + 70*(-(e*x
+ d)*c + 2*c*d - b*e)^(3/2)*c^8*d*e*f - 35*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*c^7*e^2*f - 70*(-(e*x + d)*c
+ 2*c*d - b*e)^(3/2)*c^8*d^2*g + 35*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*c^7*d*e*g + 21*((e*x + d)*c - 2*c*d +
b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^7*e*f - 21*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d
- b*e)*c^7*d*g + 15*((e*x + d)*c - 2*c*d + b*e)^3*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^6*g)/c^7)/e^2 - 2/105*(84
0*c^4*d^3*e*f*arctan(sqrt(2*c*d - b*e)/sqrt(-2*c*d + b*e)) - 1260*b*c^3*d^2*e^2*f*arctan(sqrt(2*c*d - b*e)/sqr
t(-2*c*d + b*e)) + 630*b^2*c^2*d*e^3*f*arctan(sqrt(2*c*d - b*e)/sqrt(-2*c*d + b*e)) - 105*b^3*c*e^4*f*arctan(s
qrt(2*c*d - b*e)/sqrt(-2*c*d + b*e)) - 840*c^4*d^4*g*arctan(sqrt(2*c*d - b*e)/sqrt(-2*c*d + b*e)) + 1260*b*c^3
*d^3*e*g*arctan(sqrt(2*c*d - b*e)/sqrt(-2*c*d + b*e)) - 630*b^2*c^2*d^2*e^2*g*arctan(sqrt(2*c*d - b*e)/sqrt(-2
*c*d + b*e)) + 105*b^3*c*d*e^3*g*arctan(sqrt(2*c*d - b*e)/sqrt(-2*c*d + b*e)) + 644*sqrt(2*c*d - b*e)*sqrt(-2*
c*d + b*e)*c^3*d^2*e*f - 644*sqrt(2*c*d - b*e)*sqrt(-2*c*d + b*e)*b*c^2*d*e^2*f + 161*sqrt(2*c*d - b*e)*sqrt(-
2*c*d + b*e)*b^2*c*e^3*f - 764*sqrt(2*c*d - b*e)*sqrt(-2*c*d + b*e)*c^3*d^3*g + 824*sqrt(2*c*d - b*e)*sqrt(-2*
c*d + b*e)*b*c^2*d^2*e*g - 251*sqrt(2*c*d - b*e)*sqrt(-2*c*d + b*e)*b^2*c*d*e^2*g + 15*sqrt(2*c*d - b*e)*sqrt(
-2*c*d + b*e)*b^3*e^3*g)/(sqrt(-2*c*d + b*e)*c*e^2)

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\int \frac {\left (f+g\,x\right )\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^{7/2}} \,d x \]

[In]

int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2))/(d + e*x)^(7/2),x)

[Out]

int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2))/(d + e*x)^(7/2), x)